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J. I. García-García

Publications and source records attributed to J. I. García-García.

At least 19 recordsLinked to original sources
arXiv2024

On ideals of affine semigroups and affine semigroups with maximal embedding dimension

Let $S\subseteq \mathbb N^p$ be a semigroup, any $P\subseteq S$ is an ideal of $S$ if $P+S\subseteq P$, and an $I(S)$-semigroup is the affine semigroup $P\cup \{0\}$, with $P$ an ideal of $S$. We characterise the $I(S)$-semigroups and the ones that also are $\mathcal C$-semigroups. Moreover, some algorithms are provided to compute all the $I(S)$-semigroups satisfying some properties. From a family of ideals of $S$, we introduce the affine semigroups with maximal embedding dimension, characterising them and describing some families.

math.AC↗
arXiv2024

On the quotient of affine semigroups by a positive integer

This work delves into the {\it quotient of an affine semigroup by a positive integer}, exploring its intricate properties and broader implications. We unveil an {\it associated tree} that serves as a valuable tool for further analysis. Moreover, we successfully generalize several key irreducibility results, extending their applicability to the more general class of $\mathcal C$-semigroup quotients. To shed light on these concepts, we introduce the novel notion of an {\it arithmetic variety of affine semigroups}, accompanied by illuminating examples that showcase its power.

math.AC↗
arXiv2022

The complexity of a numerical semigroup

Let $S$ and $Δ$ be numerical semigroups. A numerical semigroup $S$ is an $\mathbf{I}(Δ)$-{\it semigroup} if $S\backslash \{0\}$ is an ideal of $Δ$. We will denote by $\mathcal{J}(Δ)=\{S \mid S \text{ is an $\mathbf{I}(Δ)$-semigroup} \}.$ We will say that $Δ$ is {\it an ideal extension of } $S$ if $S\in \mathcal{J}(Δ).$ In this work, we present an algorithm that allows to build all the ideal extensions of a numerical semigroup. We can recursively denote by $\mathcal{J}^0(\mathbb{N})=\mathbb{N},$ $\mathcal{J}^1(\mathbb{N})=\mathcal{J}(\mathbb{N})$ and $\mathcal{J}^{k+1}(\mathbb{N})=\mathcal{J}(\mathcal{J}^{k}(\mathbb{N}))$ for all $k\in \mathbb{N}.$ The complexity of a numerical semigroup $S$ is the minimun of the set $\{k\in \mathbb{N}\mid S \in \mathcal{J}^k(\mathbb{N})\}.$ In addition, we will give an algorithm that allows us to compute all the numerical semigroups with fixed multiplicity and complexity.

math.NT↗
arXiv2021

On the ideal of some sumset semigroups

A sumset semigroup is a non-cancellative commutative monoid obtained from the sumset of finite non-negative integer sets. In this work, an algorithm for computing the ideals associated with some sumset semigroups is provided. Using these ideals, we study some factorization properties of sumset semigroups and some additive properties of sumsets. This approach links computational commutative algebra with additive number theory.

math.NT↗
arXiv2021

Characterizing affine $\mathcal{C}$-semigroups

Let $\mathcal C \subset \mathbb N^p$ be a finitely generated integer cone and $S\subset \mathcal C$ be an affine semigroup such that the real cones generated by $\mathcal C$ and by $S$ are equal. The semigroup $S$ is called $\mathcal C$-semigroup if $\mathcal C\setminus S$ is a finite set. In this paper, we characterize the $\mathcal C$-semigroups from their minimal generating sets, and we give an algorithm to check if $S$ is a $\mathcal C$-semigroup and to compute its set of gaps. We also study the embedding dimension of $\mathcal C$-semigroups obtaining a lower bound for it, and introduce some families of $\mathcal C$-semigroups whose embedding dimension reaches our bound. In the last section, we present a method to obtain a decomposition of a $\mathcal C$-semigroup into irreducible $\mathcal C$-semigroups.

math.AC↗
arXiv2020

The Buchweitz set of a numerical semigroup

Let $A \subset {\mathbb Z}$ be a finite subset. We denote by $\mathcal{B}(A)$ the set of all integers $n \ge 2$ such that $|nA| > (2n-1)(|A|-1)$, where $nA=A+\cdots+A$ denotes the $n$-fold sumset of $A$. The motivation to consider $\mathcal{B}(A)$ stems from Buchweitz's discovery in 1980 that if a numerical semigroup $S \subseteq {\mathbb N}$ is a Weierstrass semigroup, then $\mathcal{B}({\mathbb N} \setminus S) = \emptyset$. By constructing instances where this condition fails, Buchweitz disproved a longstanding conjecture by Hurwitz (1893). In this paper, we prove that for any numerical semigroup $S \subset {\mathbb N}$ of genus $g \ge 2$, the set $\mathcal{B}({\mathbb N} \setminus S) $ is finite, of unbounded cardinality as $S$ varies.

math.CO↗
arXiv2020

On reducible non-Weierstrass semigroups

Weierstrass semigroups are well-known along the literature. We present a new family of non-Weierstrass semigroups which can be written as an intersection of Weierstrass semigroups. In addition, we provide methods for calculating non-Weierstrass semigroups with genus as large as desired.

math.AG↗
arXiv2020

Generalized strongly increasing semigroups

In this work we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and others families of semigroups and we give explicitly their set of gaps. Moreover, an algorithm to obtain all the GSI-semigroups up to a given Frobenius number is provided and the realization of positive integers as Frobenius numbers of GSI-semigroups is studied.

math.AC↗
arXiv2019

Union of sets of lengths of numerical semigroups

Let $S=\langle a_1,\ldots,a_p\rangle$ be a numerical semigroup, $s\in S$ and ${\sf z}(s)$ its set of factorizations. The set of length is denoted by ${\mathcal L}(s)=\{{\tt L}(x_1,\dots,x_p)\mid (x_1,\dots,x_p)\in{\sf Z}(s)\}$ where ${\tt L}(x_1,\dots,x_p)=x_1+\ldots+x_p$. From these definitions, the following sets can be defined ${\textsf W}(n)=\{s\in S\mid \exists x\in{\sf z}(s) \textrm{ such that } {\tt{L}}(x)=n\}$, $ν(n)=\cup_{s\in {\textsf W}(n)} {\mathcal L}(s)=\{l_1<l_2<\ldots< l_r\}$ and $Δν(n)=\{l_2-l_1,\ldots,l_r-l_{r-1}\}$. In this paper, we prove that the set $Δν(S)=\cup_{n\in{\mathbb{N}}}Δν(n)$ is almost periodic with period ${\rm lcm}(a_1,a_p)$.

math.AC↗
arXiv2019

A geometrical characterization of proportionally modular affine semigroups

A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality $f_1x_1+\cdots +f_nx_n \mod b \le g_1x_1+\cdots +g_nx_n$ where $g_1,\dots,g_n,$ $f_1,\ldots ,f_n\in \mathbb{Z}$ and $b\in\mathbb{N}$. In this work, a geometrical characterization of these semigroups is given. Moreover, some algorithms to check if a semigroup $S$ in $\mathbb{N}^n$, with $\mathbb{N}^n\setminus S$ a finite set, is a proportionally modular affine semigroup are provided by means of that geometrical approach.

math.AC↗
arXiv2019

On pseudo-Frobenius elements of submonoids of $\mathbb{N}^d$

In this paper we study those submonoids of $\mathbb{N}^d$ which a non-trivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of $\mathbb{N}^d$ and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.

math.AC↗
arXiv2017

Semigroups with fixed multiplicity and embedding dimension

Given $m\in \mathbb{N},$ a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $\{m,m+1,\ldots, 2m-1\}.$ In this work, packed numerical semigroups are used to built the set of numerical semigroups with fixed multiplicity and embedding dimension, and to create a partition in this set. Moreover, Wilf's conjecture is checked in the tree associated to some packed numerical semigroups.

math.AC↗
arXiv2016

On divisor-closed submonoids and minimal distances in finitely generated monoids

We study the lattice of divisor-closed submonoids of finitely generated cancellative commutative monoids. In case the monoid is an affine semigroup, we give a geometrical characterization of such submonoids in terms of its cone. Finally, we use our results to give an algorithm for computing $Δ^*(H)$ the set of minimal distance of $H$.

math.AC↗
arXiv2016

An extension of Wilf's conjecture to affine semigroups

Let $\CaC\subset \Q^p$ be a rational cone. An affine semigroup $S\subset \CaC$ is a $\CaC$-semigroup whenever $(\CaC\setminus S)\cap \N^p$ has only a finite number of elements. In this work, we study the tree of $\CaC$-semigroups, give a method to generate it and study their subsemigroups with minimal embedding dimension. We extend Wilf's conjecture for numerical semigroups to $\CaC$-semigroups and give some families of $\CaC$-semigroups fulfilling the extended conjecture. We also check that other conjectures on numerical semigroups seem to be also satisfied by $\CaC$-semigroups.

math.NT↗
arXiv2016

Proportionally modular affine semigroups

This work introduces a new kind of semigroup of $\N^p$ called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We prove they are finitely generated and we give an algorithm to compute their minimal generating sets. We also specialise on the case $p=2$. For this case, we provide a faster algorithm to compute their minimal system of generators and we prove they are Cohen-Macaulay and Buchsbaum. Besides, the Gorenstein property is charactized, and their (minimal) Fröbenius vectors are determinated.

math.AC↗
arXiv2014

Affine convex body semigroups and Buchsbaum rings

In this work, new families of Buchsbaum rings, developed by mean of convex body semigroups, are presented. We characterize Buchsbaum circle and convex polygonal semigroups and describe algorithmic methods to check these characterizations.

math.AC↗
arXiv2014

Computation of Delta sets of numerical monoids

Let $\{a_1,\dots,a_p\}$ be the minimal generating set of a numerical monoid $S$. For any $s\in S$, its Delta set is defined by $Δ(s)=\{l_{i}-l_{i-1}|i=2,\dots,k\}$ where $\{l_1<\dots<l_k\}$ is the set $\{\sum_{i=1}^px_i\,|\, s=\sum_{i=1}^px_ia_i \textrm{ and } x_i\in \N \textrm{ for all }i\}.$ The Delta set of $S$, denoted by $Δ(S)$, is the union of all the sets $Δ(s)$ with $s\in S.$ As proved in [S.T. Chapman, R. Hoyer, and N. Kaplan. Delta sets of numerical monoids are eventually periodic. Aequationes Math. 77 (2009), no. 3, 273--279], there exists a bound $N$ such that $Δ(S)$ is the union of the sets $Δ(s)$ with $s\in S$ and $s<N$. In this work, by using geometrical tools, we obtain a sharpened bound and we give an algorithm to compute $Δ(S)$ from the factorizations of only $a_1$ elements.

math.AC↗